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Chapter 7: Problem 98

Simplify. $$\frac{\sqrt{h k}}{\sqrt{h}+3 \sqrt{k}}$$

Short Answer

Expert verified
Initially complex, finally simplified.

Step by step solution

01

Identify like terms

Observe that \(\frac{\text{numerator}}{\text{denominator}}\). Both the numerator and the denominator have square roots.
02

Rationalize the denominator

To simplify this, we need to eliminate the square roots in the denominator by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\frac{\text{numerator}}{\text{denominator}}\) is \(\frac{\text{numerator}}{\text{denominator with changed second term sign}}\), in this case: \( \frac{\text{numerator}}{(\text{denominator}-3k)/(denominator)} \) .
03

Apply the FOIL method to the denominator

Multiply out the denominator, using the FOIL method. We get: \(\frac{\text{numerator}}{\text{denominator}}\).
04

Simplify the fraction

Finally, combine and simplify where possible.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

simplifying fractions
Fractions may often look complicated, but by following some rules, we can make them much simpler.
One way to simplify a fraction is to make sure that both the numerator (the top part) and the denominator (the bottom part) have no common factors other than 1.
For example, in the fraction \(\frac{10}{20}\), both 10 and 20 can be divided by 10 to simplify it to \(\frac{1}{2}\).

In the exercise, we have a square root in both the numerator and the denominator: \(\frac{\sqrt{hk}}{\sqrt{h} + 3\sqrt{k}}\).
Fractions with square roots can be simplified using rationalization.
square roots
A square root of a number is a value that, when multiplied by itself, gives the original number.
For instance, the square root of 9 is 3, because 3 * 3 = 9.
Square roots are commonly denoted using the radical symbol \(\sqrt{ }\).
In algebra, we often encounter expressions like \(\sqrt{h}\) where
FOIL method
The FOIL method is a tool used to multiply two binomials, which are expressions consisting of two terms.
FOIL stands for First, Outer, Inner, Last:
  • First: Multiply the first terms in each binomial.
  • Outer: Multiply the outer terms in each binomial.
  • Inner: Multiply the inner terms in each binomial.
  • Last: Multiply the last terms in each binomial.
For example, to multiply \( (a+b)(c+d) \):
  • First: a*c
  • Outer: a*d
  • Inner: b*c
  • Last: b*d
For the given exercise, we apply the FOIL method to \( ( \sqrt{h} + 3\sqrt{k} )( \sqrt{h} - 3\sqrt{k}) \) to rationalize the denominator.

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Most popular questions from this chapter

Evaluate expression. \((-32)^{1 / 5}\)

Use a scientific calculator with a power key ( \(x^{y}\) ) to find the decimal value of each expression. Round approximate answers to four decimal places. \(2^{1 / 3}\)

Use the rules of exponents to simplify expression. \(\frac{8^{1 / 3}}{8^{2 / 3}}\)

Solve each problem by writing an equation and solving it. Find the exact answer and simplify it using the rules for radicals. If the sides of a rectangle are 30 feet and 40 feet in length, find the length of the diagonal of the rectangle.

Use the rules of exponents to simplify expression. \(\left(5^{4}\right)^{1 / 2}\)
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